If `log_(e) (x^(4) y)` = a and `log_(e) (x^(2) y^(2)) = b,` then `log_(e) sqrt(y)` in terms of a and b is

Options

A. `(2 b + a)/(2 )`
B. `( 2 b - a)/(7)`
C. `(2 b - a)/(6)`
D. none of these

Answer

To solve the problem, we need to express \( \log_e \sqrt{y} \) in terms of \( a \) and \( b \). We start with the given equations:

1. \( \log_e (x^4 y) = a \)
2. \( \log_e (x^2 y^2) = b \)

### Step 1: Rewrite the logarithmic equations

From the first equation:
\[ \log_e (x^4 y) = \log_e (x^4) + \log_e (y) = 4 \log_e (x) + \log_e (y) = a \]

From the second equation:
\[ \log_e (x^2 y^2) = \log_e (x^2) + \log_e (y^2) = 2 \log_e (x) + 2 \log_e (y) = b \]

### Step 2: Express \( \log_e (y) \) in terms of \( \log_e (x) \)

From the second equation, we can express \( \log_e (y) \) in terms of \( \log_e (x) \):
\[ 2 \log_e (x) + 2 \log_e (y) = b \implies \log_e (x) + \log_e (y) = \frac{b}{2} \]
Thus,
\[ \log_e (y) = \frac{b}{2} - \log_e (x) \]

### Step 3: Substitute \( \log_e (y) \) into the first equation

Substituting \( \log_e (y) \) into the first equation:
\[ 4 \log_e (x) + \left( \frac{b}{2} - \log_e (x) \right) = a \]
This simplifies to:
\[ 4 \log_e (x) - \log_e (x) + \frac{b}{2} = a \]
\[ 3 \log_e (x) + \frac{b}{2} = a \]

### Step 4: Solve for \( \log_e (x) \)

Rearranging gives:
\[ 3 \log_e (x) = a - \frac{b}{2} \]
\[ \log_e (x) = \frac{a - \frac{b}{2}}{3} \]

### Step 5: Substitute back to find \( \log_e (y) \)

Now substitute \( \log_e (x) \) back into the expression for \( \log_e (y) \):
\[ \log_e (y) = \frac{b}{2} - \frac{a - \frac{b}{2}}{3} \]
To simplify this, we find a common denominator:
\[ \log_e (y) = \frac{3b}{6} - \frac{2(a - \frac{b}{2})}{6} \]
\[ = \frac{3b - 2a + b}{6} = \frac{4b - 2a}{6} = \frac{2b - a}{3} \]

### Step 6: Find \( \log_e \sqrt{y} \)

Using the property of logarithms:
\[ \log_e \sqrt{y} = \log_e (y^{1/2}) = \frac{1}{2} \log_e (y) \]
Substituting our expression for \( \log_e (y) \):
\[ \log_e \sqrt{y} = \frac{1}{2} \cdot \frac{2b - a}{3} = \frac{2b - a}{6} \]

Thus, the final answer is:
\[ \log_e \sqrt{y} = \frac{2b - a}{6} \]